If \(y=\operatorname{Tan}^{-1} \sqrt{x^2-1}+\operatorname{Sinh}^{-1} \sqrt{x^2-1}, x>1\), then \(\frac{d y}{d x}=\)

If it is mentioned that for a curve passing through \((3,4)\), the slope of the curve at any point is the reciprocal, of twice the ordinate of that point, then that curve is a

The length of the chord joining points \((4 \cos \theta, 4 \sin \theta)\) and \(\left[4 \cos \left(\theta+60^{\circ}\right)\right.\), \(\left.4 \sin \left(\theta+60^{\circ}\right)\right]\) on the circle \(x^2+y^2=16\) is

If \(a>0\) and the coefficient of \(x^2\) in the expansion of \(\left(a x^3+\frac{c}{x}\right)^6\) is 60, then \(a c^2=\)

The function \(f: R \rightarrow R\) is defined by \(f(x)=\cos ^2 x+\sin ^4 x\) for \(x \in R\), then \(f(R)\) is equal to

The number of rational terms in the expansion of ${\left(3^{\frac{1}{4}}+7^{\frac{1}{6}}\right)}^{144}$ is

If \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\), \(\mathbf{n}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\) and \(\theta\) is the angle between \(\mathbf{c}\) and \(\mathbf{n}\) then \(\sin \theta=\)

If $X$ is a Poisson variate such that $2P(X=1)=5P(X=5)+2P(X=3),$ then the standard deviation of $X$ is

If \(I_n=\int \sin ^n x d x\), then \(n I_n-(n-1) I_{n-2}\) equals

Incorrect matching amongst the following is (according to geometry of molecules)

Which of the following statement is correct for fcc lattice?

A \(0.5 \mathrm{~kg}\) block of brass (density \(=8 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\) ) is suspended from a string. What is the tension in the string if the block is completely immersed in water? \(\left(g=10 \mathrm{~ms}^{-2}\right)\)

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)